Lemma 8 (Corollary 3.1 in [1]}) 

Suppose \(X_1,\dots ,X_n\)  are identically distributed \((K^{\prime },\alpha )\)  sub-Weibull random variables. Then, for some absolute constant \(c > 0\) , for all \(s \ge n c K^{\prime }\) , we have
\(\mathbb {P}\left(*{\sum _{i=1}^n X_i} \ge s \right) \le \exp \left(-\left(\frac{s}{n c K^{\prime }}\right)^{1/\alpha } \right)\) 

